Theorem sseqtrdi 3145

Description: A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)

Hypotheses

Ref	Expression
sseqtrdi.1	|- ( ph -> A C_ B )
sseqtrdi.2	|- B = C

Assertion

Ref	Expression
sseqtrdi	|- ( ph -> A C_ C )

Proof of Theorem sseqtrdi

Step	Hyp	Ref	Expression
1		sseqtrdi.1	. 2 |- ( ph -> A C_ B )
2		sseqtrdi.2	. . 3 |- B = C
3	2	sseq2i 3124	. 2 |- ( A C_ B <-> A C_ C )
4	1, 3	sylib 121	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1331   C_ wss 3071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-in 3077  df-ss 3084

This theorem is referenced by:  sseqtrrdi  3146  onintonm  4433  relrelss  5065  iotanul  5103  foimacnv  5385  cauappcvgprlemladdru  7464  zsumdc  11153  fsum3cvg3  11165  distop  12254  cnptoprest  12408  pwle2  13193